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To find the acute angle between the lines given by the equations \( x + 5y = 11 \) and \( 5x - y = 3 \), we will first express both equations in the slope-intercept form \( y = mx + c \), where \( m \) is the slope.

### Step 1: Convert both equations to slope-intercept form

1. **First equation**: \( x + 5y = 11 \)
   \[
   5y = -x + 11
   \]
   \[
   y = -\frac{1}{5}x + \frac{11}{5}
   \]
   So, the slope of the first line, \( m_1 = -\frac{1}{5} \).

2. **Second equation**: \( 5x - y = 3 \)
   \[
   -y = -5x + 3
   \]
   \[
   y = 5x - 3
   \]
   So, the slope of the second line, \( m_2 = 5 \).

### Step 2: Formula for the angle between two lines

The formula to find the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:
\[
\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|
\]
Substituting \( m_1 = -\frac{1}{5} \) and \( m_2 = 5 \) into the formula:

\[
\tan \theta = \left| \frac{5 - \left(-\frac{1}{5}\right)}{1 + \left(-\frac{1}{5}\right)(5)} \right|
\]
\[
\tan \theta = \left| \frac{5 + \frac{1}{5}}{1 - 1} \right|
\]
This results in a division by zero, indicating that the lines are perpendicular, so the angle between them is 90 degrees. However, the problem asks for the acute angle. Since perpendicular lines have an angle of \( 90^\circ \), the acute angle between them would be \( 90^\circ \).

Would you like further details or have any questions?

### Relative Questions:
1. How do you find the angle between two lines using their slopes?
2. What is the meaning of the slope-intercept form of a line?
3. Can two lines with zero slopes ever have an angle between them?
4. What does it mean if the denominator in the tangent angle formula is zero?
5. How would the solution change if the lines were not perpendicular?

### Tip:
Always check the slopes of lines first; if their product equals \(-1\), the lines are perpendicular.

Find the acute angles between the lines x + 5y = 11 and 5x - y = 3.

Discover how to calculate the acute angle between two lines using slope-intercept form and the angle formula. This example uses the lines x + 5y = 11 and 5x - y = 3, suitable for Grades 9-12.

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